Adam: full update rule

The problem, from first principles

Adam: bias-corrected moment estimates built the two pieces, m (a bias-corrected running average of the gradient, exactly SGD + Momentum's velocity) and v (a bias-corrected running average of the SQUARED gradient, a new idea: tracking gradient MAGNITUDE separately from direction). This question assembles them into Adam's actual parameter update, and the way they combine is genuinely clever: m_hat decides WHICH WAY to step (momentum's own job), while sqrt(v_hat) decides HOW FAR, dividing the step down for any parameter whose gradients have consistently been large, and allowing a relatively bigger step for a parameter whose gradients have consistently been small.

This gives every parameter, effectively, its OWN adaptive learning rate, computed automatically from its own gradient history, rather than the single, uniform lr every parameter in SGD/SGD + Momentum shares.

From theory to code

Theory updates and bias-corrects both moments (reusing Adam: bias-corrected moment estimates's own functions directly), then combines them into one update: step opposite the corrected mean gradient, scaled by the learning rate and divided by the corrected root-mean-square gradient magnitude (plus a tiny eps for numerical safety).

Implement adam_step(params, grads, m_list, v_list, t, lr=0.001, beta1=0.9, beta2=0.999, eps=1e-8) against that reasoning. The signature and docstring are already in the editor.

Constraints

• t starts at 1 (1-indexed), matching bias_correct's own convention.
• Both m_list and v_list must be threaded through to the caller's next step (same pattern SGD + Momentum's velocities used).
• lr defaults to 0.001, matching real Adam's own much-smaller-than-SGD default (the adaptive per-parameter scaling means a smaller base lr is typically appropriate).

Hints

Open one at a time. Each gives away a little more than the last.